Effect of Electrolyte Concentration and Pore Size on Ion Current Rectification Inversion

A thorough understanding of nanoscale transport properties is vital for the development and optimization of nanopore sensors. The thickness of the electrical double layers (EDLs) at the internal walls of a nanopore, as well as the dimensions of the nanopore itself, plays a crucial role in determining transport properties. Herein, we demonstrate the effect of the electrolyte concentration, which is inversely proportional to the EDL thickness, and the effect of pore size, which controls the extent of the electrical double layer overlap, on the ion current rectification phenomenon observed for conical nanopores. Experimental and numerical results showed that as the electrolyte concentration is decreased, the rectification ratio reaches a maximum, then decreases, and eventually inverts below unity. We also show that as the pore size is decreased, the rectification maximum and the inversion take place at higher electrolyte concentrations. Numerical investigations revealed that both phenomena occur due to the shifting of ion enrichment distributions within the nanopore as the electrolyte concentration or the pore size is varied.


Meshing and Boundary Conditions
The meshing is constructed to be small near the tip opening and near the double layer ( Figure S1b), while larger meshing is used in the bulk solution and in the further interior of the nanopipette ( Figure S2a). The conical region, the double layers, and the space immediately outside the pore are further segregated into separate domains to allow better control of the mesh size. The mesh consists of triangular elements only.
Since the tip size is changed between computations, it was important to adapt the mesh size at the tip to ensure that the meshing stays sufficiently fine without the computations becoming too costly. Mesh refinement studies were carried out and no change in rectification was observed upon further reduction of the mesh size, indicating that the mesh is sufficient to resolve the variations in the field variables solved for.
The bulk solution was constructed to be circular so that all bulk solution boundary elements are an equal distance from the tip mouth. Further increases in the length of the nanopipette and the radius of the bulk solutions were found to have no effect on the results of the model. Boundary conditions were applied as shown in Figure S2. A constant concentration boundary was applied to the bulk solution boundaries and to the interior solution of the nanopipette, a potential boundary condition (+0.6 V or -0.6 V) was applied to the interior of the pipette and a ground boundary condition to the bulk solution. No slip conditions were applied to both the interior and exterior nanopipette walls, and no pressure gradient was applied between the interior and exterior solutions. Lastly, no flux and surface charge boundary conditions of = 1 mC m -2 , were applied to both the inside and outside nanopipette walls.

Figure S1
The FEM geometry shown for the (a) whole model and (b) the tip region.

Figure S2
Boundary conditions applied in the FEM model.

The Effect of Electroosmotic Flow
Finite Element Analysis was carried out both with and without the Navier-Stokes equations which include the electroosmotic body force. As visible on Figure S3, the electroosmotic flow has a significant effect for the larger pores where the magnitude of the rectification ratio at the rectification maximum increases and where the peak maximum shifts to slightly lower concentrations. This effect is negligible for the smaller 6 nm pore, where the two curves are superimposed.

Rectification Ratio
Concentration mmol dm -3 251 nm with NS 251 nm no NS 6 nm with NS 6 nm no NS

Figure S3
The effect of including the Navier-Stokes equations for the largest and the smallest pores. Figure S4 shows the cation transference number of the pore as a function of electrolyte concentration.  Figure S5 shows the cation and anion traces associated with Figure 3 in the main paper. It shows how the cation and anion traces become significantly different as the EDL length increases.  Figure S6 shows that the individual cation and anion enrichment also shifts as a function of electrolyte concentration. At the negative potential, both the cation and anion enrichment peaks shift further outside the pore as the electrolyte concentration is lowered. On the other hand, at the positive potential, the cation enrichment peak shifts inside the pore, while the anion enrichment peak shifts outside as the electrolyte concentration is decreased.  Figure S7 shows that the cation and anion ion enrichment traces shift with the pore size. The cation enrichment peaks shift outside the pipette as the pore size is decreased at both the positive and negative potentials. On the other hand, the anion enrichment peak shifts further inside the pipette as the pore size is decreased at both the positive and negative potentials.

Second inversion of rectification
A second inversion of rectification, similar to that reported by Momotenko et al., was also observed by us numerically, however, the experimental observation of this second rectification was beyond our measurement capabilities. The second inversion was predicted to occur at an electrolyte concentration of around 0.0001 mM for the 6 nm pores, however, the noise associated with the measurement of the current-voltage curves at these concentrations would lead to a large uncertainty in the extracted RR which would not allow the reliable experimental observation of the second rectification. Furthermore, at such low concentrations, the uncertainty in the concentration of the prepared solutions also becomes significant.

Sample Current-Voltage Curves
A Biologic SP-200 potentiostat was used for collecting the current-voltage curves. Ultra-low current option at a high-speed scan is employed; a filter bandwidth of 50 kHz is used during data collection. Noise is further reduced by numerically filtering the data after their acquisition using a moving average filter with a windows size of 11 points in EC-lab V11.34. The parameter settings for the recorded current-voltage curves includes 3 scans taken at a scan rate of 0.1 V/s within a potential window of -0.6 to +0.6 V. Figure S8 shows samples of the current-voltage curves for each pore size at the electrolyte concentration where maximum rectification was observed. It is important to point out that the noise is the smallest for the 40 nm pore at maximum rectification. This arises since the noise is proportional to the pore size (larger pores carry larger currents) but inversely proportional to the electrolyte concentration (smaller electrolyte concentrations lead to smaller current magnitudes). Since larger pores have their rectification maximum at lower electrolyte concentrations, the noise at the rectification maximum is minimized for the 40 nm pore.

Description Value
Electric potential volt

Description Value
Apply reaction terms on All physics (symmetric)

Size (size)
Cylinder

Description Value
Maximum element size 0.5e-9 Minimum element size 1.81E-9 Minimum element size Off